Question: All of the 5th grade teachers and students from Oak went on a field trip to an art museum. Tickets were $$5.50$ each for teachers and $$4.00$ each for students, and the group paid $$46.00$ in total. The next month, the same group visited a science museum where the tickets cost $$16.50$ each for teachers and $$10.50$ each for students, and the group paid $$129.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+4y = 46}$ ${16.5x+10.5y = 129}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-16.5x-12y = -138}$ ${16.5x+10.5y = 129}$ Add the top and bottom equations together. $ -1.5y = -9 $ $ y = \dfrac{-9}{-1.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {5.5x+4y = 46}$ to find $x$ ${5.5x + 4}{(6)}{= 46}$ $5.5x+24 = 46$ $5.5x = 22$ $x = \dfrac{22}{5.5}$ ${x = 4}$ You can also plug ${y = 6}$ into $ {16.5x+10.5y = 129}$ and get the same answer for $x$ ${16.5x + 10.5}{(6)}{= 129}$ ${x = 4}$ There were $4$ teachers and $6$ students on the field trips.